Problem: Multiply the following complex numbers, marked as blue dots on the graph: $( e^{3\pi i / 4}) \cdot (3 e^{5\pi i / 3})$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $ e^{3\pi i / 4}$ ) has angle $\frac{3}{4}\pi$ and radius $1$ The second number ( $3 e^{5\pi i / 3}$ ) has angle $\frac{5}{3}\pi$ and radius $3$ The radius of the result will be $1 \cdot 3$ , which is $3$ The sum of the angles is $\frac{3}{4}\pi + \frac{5}{3}\pi = \frac{29}{12}\pi$ The angle $\frac{29}{12}\pi$ is more than $2 \pi$ . A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $\frac{29}{12}\pi - 2 \pi = \frac{5}{12}\pi$ The radius of the result is $3$ and the angle of the result is $\frac{5}{12}\pi$.